Table Of ContentVirginia Polytechnic Institute
And State University
The Charles E. Via, Jr.
Department of Civil Engineering
CENTER FOR
GEOTECHNICAL PRACTICE AND RESEARCH
Beginner’s Guide For Geotechnical
Finite Element Analyses
by
Nathaniel Bradley
and
Daniel R. VandenBerge
Report of a study performed by the Virginia Tech Center for
Geotechnical Practice and Research
May 2015
CGPR # 82
Center for Geotechnical Practice and Research
200 Patton Hall, Vi rginia Tech
Blacksburg, Virginia 24060
TABLE OF CONTENTS
Acknowledgement ....................................................................................................... 1
Chapter 1 – Introduction ............................................................................................. 2
Chapter 2 - Fundamentals .......................................................................................... 5
Finite Element Programs ......................................................................................... 7
Chapter 3 - STEADY-STATE SEEPAGE .................................................................... 8
Governing Principles ............................................................................................... 8
Finite Element Mesh Design .................................................................................... 8
Boundary Conditions ............................................................................................. 10
Free Surface Flow Problems ................................................................................. 11
Element Types ...................................................................................................... 12
Mesh Refinement .................................................................................................. 12
Transient Seepage Analysis .................................................................................. 13
Capabilities of Commercially Available Finite Element Programs ......................... 14
Examples ............................................................................................................... 15
Example 1: Steady-State Seepage under a Gravity Dam .................................. 15
Parametric Study ............................................................................................... 24
Example 2: Steady-State Seepage through a Compacted Silty Sand
Embankment .......................................................................................................... 25
Validation ............................................................................................................... 33
Chapter 4 – STRESSES AND MOVEMENTS ........................................................... 35
Principal Variables ................................................................................................. 35
Shape Functions for Load-Deformation Problems ................................................. 36
Stress-Strain and Strength Parameters for Soils ................................................... 36
Linear Elastic Stress-Strain ................................................................................ 37
Hyperbolic Stress-Strain and Strength ............................................................... 38
Selection of Hyperbolic Parameters ................................................................... 41
Initial Stresses and Displacements ........................................................................ 43
Drained vs. Undrained Conditions ..................................................................... 43
Initial Stresses in Phase2 ................................................................................... 44
Modeling Construction Sequence .......................................................................... 45
Capabilities of Modern Finite Element Programs .................................................. 46
Examples ............................................................................................................... 46
Example 3: Embankment Constructed on Stiff Foundation ................................ 46
Example 4: Embankment Constructed on Clay Foundation. .............................. 54
Example 5: Excavation in Clay .......................................................................... 62
Chapter 5 – Stability Analyses .................................................................................. 76
Strength Reduction Analysis ................................................................................. 76
Example 6 .......................................................................................................... 77
Advantages and Disadvantages ............................................................................ 79
Rules of Thumb and Guidelines ............................................................................ 80
Chapter 6 – Guidelines and Lessons Learned .......................................................... 81
Fundamentals ........................................................................................................ 81
Seepage ................................................................................................................ 81
Modeling ............................................................................................................ 81
Validation ........................................................................................................... 81
Load Deformation .................................................................................................. 82
Stress-Strain Properties ..................................................................................... 82
Stress-Strain Modeling ....................................................................................... 82
Validation ........................................................................................................... 83
Stability Analysis ................................................................................................... 83
References ................................................................................................................ 84
Appendix A Staged Embankment Constructions in Phase2 ................................... 86
Analysis Steps ....................................................................................................... 86
Appendix B Initial Stresses in Level Ground in Phase2 .......................................... 91
Analysis Steps ....................................................................................................... 91
ACKNOWLEDGEMENT
This Beginner’s Guide for Finite Element Analyses was written at the suggestion of
the CGPR members at the 2014 Annual Meeting, and the CGPR provided most of the
financial support for the effort. Virginia Tech provided financial support during the final
phases of the project through Dan VandenBerge’s post-doctoral position. All of this
support is gratefully acknowledged.
The authors thank Professor Mike Duncan for his guidance, encouragement, and
contributions to this project and the development of this report. His insights and
practical experience in the application of the finite element method were essential to the
report’s success.
1
CHAPTER 1 – INTRODUCTION
This manual is intended for the use of geotechnical engineers and geotechnical
engineering graduate students who understand the behavior of soil and conventional
methods for analyzing seepage, settlement and stability, but who have not performed
finite element analyses.
With the objective of removing the “black box” aspect of finite element analyses,
Chapter 2 is devoted to the fundamentals of the method, so that the user will
understand concepts such as nodal points, elements, and boundary conditions, and the
sets of equations that are formed and solved in the process of performing a finite
element analysis. While it is not necessary that the user be able to derive these, an
understanding of these concepts is important for performing and interpreting the results
of finite element analyses.
Chapter 3 is devoted to seepage problems, both confined flow and free surface
steady state flow. Finite element analyses offer great advantages for analysis of
seepage problems, through the ability to perform parametric studies at the click of a
button. Hydraulic conductivity can never be evaluated precisely, and the ability to study
the effects of possible variations in hydraulic conductivity and anisotropy is essential for
thorough analyses of seepage through soil. Two examples illustrate applications to
seepage beneath and through dams.
Chapter 4 is devoted to problems of stresses and movements in embankments and
around excavations, and the use of excavation bracing to control these movements.
Deformation analyses are more complex than seepage analyses, both in terms of the
soil properties involved and the techniques necessary to model nonlinear stress-strain
behavior in soils. Three examples illustrate deformations of embankments during
construction and movements around braced and unbraced excavations in clay.
Appendices A and B are devoted to important details of staged analyses to model
successive constructions steps, and the procedures used in the computer program
Phase2 to establish values of initial stresses in the ground prior to embankment
construction or excavation.
The balance of this chapter is devoted to summaries of papers and reports that
illustrate practical uses of the finite element analyses, to illustrate the wide range of
practical uses of the method. These papers and reports include:
The Behavior of Embankment Dams: Kulhawy was one of the first to perform
analyses using nonlinear stress-strain behavior of soil in finite element analyses.
2
The analyses he conducted demonstrated that displacements within embankment
dams could be predicted with reasonable accuracy (Kulhawy 1969).
Earth Pressures on and Deflections of Reinforced Concrete Structures: Clough and
Duncan used finite element analyses to compute earth pressures and deflections of
the reinforced concrete walls of Port Allen Lock, a reinforced concrete structure 75
feet wide by 75 feet high by 1,100 feet long. They compared the results of their
analyses with data from field instruments to determine the ability of analyses to
simulate field behavior (Clough and Duncan 1969).
Reinforced Embankments: Duncan and Schaefer conducted finite element analyses
of a reinforced embankment constructed over soft clay and peat. The analyses
demonstrated the feasibility of constructing an embankment over weak soil using a
single layer of reinforcement near the base of the embankment (Duncan and
Schaefer 1988).
Reinforced Soil Walls: The stability of reinforced soil walls was evaluated using the
finite element method for several different types of reinforcement by Adib, Mitchell,
and Christopher (Adib et al. 1990).
Urban Excavations: A simulation of the construction of an underground parking
garage at Post Office Square in Boston was completed by Whittle, Hashash and
Whitman. They found that it is possible to make useful predictions of deformations
occurring during top-down construction of an excavation using finite element
analyses (Whittle et al. 1993).
Lateral Spreading of Unreinforced Embankments: Crawford et al. performed a finite
element analysis on an embankment to study the influence of lateral spreading on
consolidation settlement. A two-dimensional consolidation analysis compared better
with field measurements than a one-dimensional consolidation analysis (Crawford et
al. 1994).
Stability of Slopes and Dams: Finite element analyses for the assessment of slope
stability have been conducted by Griffiths and Lane. By comparison with traditional
limit equilibrium techniques, they demonstrate that the finite element method can be
a useful tool for evaluating stability of slopes (Griffiths and Lane 1999). Hammah et
al. (2006) performed slope stability analyses using finite element software. Analyses
were performed on a broad range of slopes and the results compared well with limit
equilibrium methods.
Pile-Supported Bridge Abutments: Ellis and Springman analyzed a bridge abutment
supported by piles using two-dimensional finite element analyses. Aspects of the
structure that influenced its behavior in three dimensions, such as the piles and
3
vertical drains, were successfully implemented into the two-dimensional analysis
(Ellis and Springman 2001).
Soil-Bentonite Cutoff Walls: Baxter and Filz developed a finite element model to
simulate construction of soil-bentonite cutoff walls. They used this model to predict
the ground deformations adjacent to the cutoff wall (Baxter and Filz 2007).
Retaining Wall Failures: The failure of a segmental retaining wall was studied by
Hossain and his colleagues using finite element analysis. Analyses were able to
replicate the behavior of the wall prior to failure, and they provided insight regarding
the causes of the failure (Hossain et al. 2009).
4
CHAPTER 2 - FUNDAMENTALS
The finite element method involves discretization of the physical body being
analyzed, dividing the geometry into small areas where properties can be assumed to
be uniform. An example of the discretization of a rectangular body is shown in Figure
2.1. The smaller areas into which the larger body is divided are the “finite elements.”
The elements are connected to each other only at their corners, or “nodes.” Elements
always have nodes at their corners. Some more complex elements also have nodes
along their sides. Elements used for two-dimensional analyses in geotechnical
engineering are usually triangles or quadrilaterals. Some triangular elements may be
right triangles, and some quadrilateral elements may be rectangles, but any triangular or
quadrilateral shape can be used. Triangular elements normally have either 3 or 6
nodes, while quadrilaterals usually have either 4 or 8 nodes. 6-node triangles and 8-
node quadrilaterals have nodes along their sides.
Nodes
Typical Element
Elements
Figure 2.1: Discretization of a Rectangular Body into Finite Elements.
After discretization of the problem geometry into elements, the next step in the finite
element method is definition of the primary variable (or the primary unknown). For
steady-state seepage problems, the primary variable is hydraulic head (h). For load-
deformation problems, the primary variable is displacement (u). Element characteristics
called “shape functions” determine how the value of the primary variable varies within
each element. Finite element computer programs provide various types of elements
and suggestions for selecting elements for various purposes. Element types for steady-
state seepage problems are discussed in Chapter 3. Element types for load-
deformation problems are discussed in Chapter 4.
5
After the geometry has been subdivided into elements connected at nodes, the
computer program forms a set of equations which relate the value of the primary
unknown at each node to another variable at each node. For steady-state seepage
problems, the second variable is the flow at the node. So, for steady-state seepage
problems the heads at the nodes are related to the flows at the nodes. The equations
are based on Darcy’s Law and values of permeability for seepage problems. For load-
deformation problems, the second variable is the externally applied force at the node.
The displacements at the nodes are related to the forces at the nodes, using the
requirements of equilibrium and the stress-strain properties.
Once the governing equations have been formed by the computer program, they are
combined to form a global set of equations that describe the properties of the entire
system. Heads and flows are related by a “transmissibility” matrix, while displacements
and forces are related by a “stiffness” matrix. Equations 1 and 2 express the general
form of these equations. These matrices satisfy the condition that the entire system
must remain continuous. They also describe the material properties and boundary
conditions of the problem.
For seepage problems:
Vector of
Transmissibility Nodal Vector of
(1)
Matrix Hydraulic Nodal Flows
Heads
For load-deformation problems:
Vector
Stiffness of Vector of
(2)
Matrix Nodal Nodal Forces
Displacements
These global equations are solved by the computer program to obtain the value of
the primary variable at each nodal point. Using the values of the primary variable, the
values of other unknowns (such as pore pressures or strains) can be calculated for
each nodal point.
When compared with other techniques, the finite element method has several
advantages. First, problems involving non-homogeneous earth masses are no more
difficult to analyze than problems involving masses with homogeneous properties. The
6
finite element method can account for the variation of properties from one element to
another. Complex boundary conditions can also be accommodated readily. Third,
limitations in the user’s capabilities in mathematical analysis do not prevent use of the
finite element method. As long as the user has a thorough understanding of the
problem, including the geology, site conditions, properties of soil and rock, and
construction processes, the finite element method can produce useful and accurate
results.
Finite Element Programs
The examples in this report were analyzed using the computer programs Slide v.6.0
and Phase2 v.9.0 (recently renamed RS2). These programs are available for use on
academic projects through the Rocscience Education Program. Slide is a 2-
dimensional slope stability program that contains a module for finite element seepage
analyses. The Rocscience finite element program Phase2 can be used for both
seepage and load-deformation analyses.
Full versions of the Rocscience programs used to prepare the examples in this report
can be downloaded from the Rocscience website (http://www.rocscience.com/
downloads/trial_versions).
7
Description:This manual is intended for the use of geotechnical engineers and geotechnical engineering graduate students who understand the behavior of soil and conventional methods for analyzing seepage, settlement and stability, but who have not performed finite element analyses.
